Question: Given $ \overrightarrow{OA}\perp\overrightarrow{OC}$, $ m \angle BOC = 7x - 102$, and $ m \angle AOB = 2x + 39$, find $m\angle AOB$. $O$ $A$ $C$ $B$
From the diagram, we see that together ${\angle AOB}$ and ${\angle BOC}$ form ${\angle AOC}$ , so $ {m\angle AOB} + {m\angle BOC} = {m\angle AOC}$ Since we are given that $\overrightarrow{OA}\perp\overrightarrow{OC}$ , we know ${m\angle AOC = 90}$ Substitute in the expressions that were given for each measure: $ {2x + 39} + {7x - 102} = {90}$ Combine like terms: $ 9x - 63 = 90$ Add $63$ to both sides: $ 9x = 153$ Divide both sides by $9$ to find $x$ $ x = 17$ Substitute $17$ for $x$ in the expression that was given for $m\angle AOB$ $ m\angle AOB = 2({17}) + 39$ Simplify: $ {m\angle AOB = 34 + 39}$ So ${m\angle AOB = 73}$.